We are trying to find a way to solve the following problem.
We need to find a way to formalise and then solve for a minimum a very large (~100 million variables) simultaneous equation with the following characteristics.
We have a large and unknown number of sets each with an unknown number of variables in each set where each variable must take an integer value (either +ve or -ve) which is unique within the set. We know that there are approximately 100 million variables in total, and each set will have at least 1 variable.
We then have ~1 billion relationships between variables with the structure [x-y] and are trying to minimise the sum of these relationships. Some variables appear only once in the equation whereas others appear thousands of times. Relationships can exist both between variables in the same set and between sets. We have some computation resources to throw at the problem but are unsure how to approach it and if it’s tractable.
Happy to provide further information and any help is hugely appreciated!